INTRODUCTION TO FINITE COXETER GROUPS, IWAHORI–HECKE ALGEBRAS AND COMPUTER PROGRAMS.

GÖTZ PFEIFFER

Abstract. The aim of this series of lectures is to give an elementary introduction into the theory of finite Coxeter Groups and Iwahori–Hecke algebras, leading up to the concept of the character table of an Iwahori–Hecke algebra and methods for its calculation. Special emphasis will be given to algorithms and explicit calculations in concrete situations.

1. Preliminaries: Finite Coxeter Groups, Basic Algorithms.

Motivation: Iwahori–Hecke algebras as Endomorphism Rings; finite Coxeter groups • as reflection groups, • by generators and relations, • from their classification; how to represent them on a computer; Matsumoto’s Theorem; the longest element.

2. (Lesser Known Properties of) Parabolic Subgroups.

Distinguished coset representatives; Deodhar’s Lemma; the coset poset; the descent algebra; conjugacy of parabolic subgroups; the parabolic table of marks.

3. Conjugacy Classes, Special Elements, Braid Monoids.

Cuspidal classes; parameterization of the conjugacy classes; the conjugacy graph and minimal elements; the braid monoid and good elements.

4. Irreducible Representations.

Exterior powers of the reflection representation; irreducible representations of the classical types; induced characters.

5. Iwahori–Hecke Algebras as Symmetric Algebras.

Trace Functions, Schur Elements, Specializations, Tits’s Deformation Theorem; Class Polynomials, the Character Table of an Iwahori–Hecke algebra, Iwahori’s Theorem.

6. Character Formulas, Generic Degrees.

Hoefsmit’s Representations, Murnaghan–Nakayama Formulas for the Classical Types.

References

[1] M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Oxford University Press, 2000.

[2] M. Geck, G. Hiss, F. Lübeck, G. Malle and G. Pfeiffer, CHEVIE — A system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), 175–210.